Full Duplex-Assisted Intercell Interference Cancellation ...welcom.buaa.edu.cn/wp-content/uploads/publications/hanshengqian/… · HAN et al.: FULL DUPLEX-ASSISTED INTERCELL INTERFERENCE

5218 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 63, NO. 12, DECEMBER 2015

Full Duplex-Assisted Intercell InterferenceCancellation in Heterogeneous NetworksShengqian Han, Member, IEEE, Chenyang Yang, Senior Member, IEEE, and Pan Chen

Abstract—This paper studies the suppression of cross-tierintercell interference (ICI) generated by a macro base station(MBS) to pico user equipments (PUEs) in heterogeneous networks(HetNets). Different from existing ICI avoidance schemes such asenhanced ICI cancellation (eICIC) and coordinated beamform-ing, which generally operate at the MBS, we propose a full duplex(FD)-assisted ICI cancellation (fICIC) scheme, which can operateat each pico BS (PBS) individually and is transparent to the MBS.The basic idea of the fICIC is to apply FD technique at the PBSsuch that the PBS can send the desired signals and forward the lis-tened cross-tier ICI simultaneously to PUEs. We first consider thenarrowband single-user case, where the MBS serves a single macroUE and each PBS serves a single PUE. We obtain the closed-formsolution of the optimal fICIC scheme, and analyze its asymptoticalperformance in ICI-dominated scenario. We then investigate thegeneral narrowband multiuser case, where both MBS and PBSsserve multiple UEs. We devise a low-complexity algorithm to opti-mize the fICIC aimed at maximizing the downlink sum rate of thePUEs subject to user fairness constraint. Finally, the generaliza-tion of the fICIC to wideband systems is investigated. Simulationsvalidate the analytical results and demonstrate the advantages ofthe fICIC on mitigating cross-tier ICI.

Index Terms—Inter-cell interference cancellation, Full duplex,Heterogeneous networks, eICIC, CoMP, OFDM.

I. INTRODUCTION

C ELLULAR systems are evolving toward heterogeneousnetworks (HetNets) with universal frequency reuse in

order to support the galloping demand of mobile wireless ser-vices [1]. By deploying low-power nodes such as micro, pico,or femto base stations (BSs) within the coverage of traditionalmacro BSs (MBSs), HetNets bring the network close to theuser equipments (UEs) to obtain the “cell-splitting” gain, andtherefore improve the area spectral efficiency. For simplicity,we refer to the low-power nodes as pico BSs (PBSs) hereinafter.

In practice, however, straightforwardly deploying PBSs inthe coverage of MBSs cannot effectively realize the promisedbenefits of HetNets because the large difference of the twotypes of BSs in transmit power makes pico UEs (PUEs) suf-fer from severe cross-tier inter-cell interference (ICI) generated

Manuscript received March 30, 2015; revised July 22, 2015; acceptedSeptember 3, 2015. Date of publication October 26, 2015; date of currentversion December 15, 2015. This work was supported in part by the NationalNatural Science Foundation of China (No. 61301084) and by the FundamentalResearch Funds for the Central Universities. The associate editor coordinatingthe review of this paper and approving it for publication was T. Q. S. Quek.

The authors are with the School of Electronics and Information Engineering,Beihang University, Beijing 100191, China (e-mail: [email protected];[email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCOMM.2015.2494046

by the MBS. Efficient ICI cancellation (ICIC) mechanisms aretherefore critical to support the HetNet deployment [1].

To mitigate the ICI, enhanced ICIC (eICIC) techniques havebeen developed in Release 10 of Long-term Evolution (LTE)[1]. In the time-domain eICIC, the MBS remains silent inthe so-called almost blank subframes (ABS), during whichthe cell-edge PUEs are served without interference. In thefrequency-domain eICIC, the MBS and PBSs schedule UEs inorthogonal frequency resources to avoid the ICI. The eICICmethods are of low complexity and easy to implement, butthey limit the performance of both macro UEs (MUEs) andPUEs since the UEs can be only served in partial time-frequency resources. Coordinated multi-point (CoMP) trans-mission is another promising technique for cross-tier ICI sup-pression, which exploits the antenna resources of the MBS [2].Considering the fact that providing high-performance backhaulfrom all PBSs to the core network may be cost prohibitive, coor-dinated beamforming (CB) based CoMP transmission, requir-ing no data sharing among the BSs, has received wide attention.CoMP-CB can be considered as a sort of spatial-domain eICIC,which enables the PUEs to be served with the whole time-frequency resources. However, the performance of CoMP-CBis limited by the number of antennas at the MBS, which is usu-ally not acceptable when the PBSs are densely deployed in thecoverage of the MBS.

Different from eICIC and CoMP-CB, both of which con-trol the transmission of the MBS in time, frequency or spatialdomain in order to generate an ICI-free environment for thetransmission between PBSs and PUEs, in this paper we striveto study the cross-tier ICI suppression scheme without theparticipation of MBSs, which therefore will not consume theresources of the MBS. Specifically, we consider using fullduplex (FD) technique to HetNets. FD communication waslong believed impossible in wireless system design due to thesevere self-interference within the same transceiver. However,the belief has been overturned recently with the tremendousprogress in self-interference cancellation [3], [4], where theplausibility of FD technique for short-range point-to-point com-munications was approved. FD techniques have been appliedto provide bi-directional communications over the same timeand frequency resources, and exhibited noticeable spectralefficiency gains over half-duplex (HD) schemes [5], [6]. FDtechniques are also applied in relay systems to improve servicecoverage, where the usage of FD avoids the waste of resourcesas in HD relay systems [7], [8]. To the best of our knowledge,leveraging FD in HetNets to suppress the cross-tier ICI has notbeen addressed in the literature.

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HAN et al.: FULL DUPLEX-ASSISTED INTERCELL INTERFERENCE CANCELLATION 5219

In this paper, we consider a HetNet consisting of a MBS andmultiple PBSs. We investigate the application of FD techniqueto mitigate the ICI generated by the MBS to the PUEs. Themain contributions are summarized as follows:

• We propose a novel ICI cancellation scheme for HetNets,named FD assisted ICI cancellation (fICIC), where thePBSs are supposed to have the FD capability of trans-mitting and receiving simultaneously. The basic idea ofthe proposed fICIC is to let the FD PBSs forward thelistened interference from the MBS to the PUEs to neu-tralize the ICI, while at the same time sending the desiredsignals. Since the ICI is mitigated at PBSs now, the pro-posed fICIC is transparent to the MBS in the sense that nochanges are needed for the transmission of the MBS. Notethat a similar usage of the FD technique was presentedin [9] for the cooperative cognitive network, where theFD secondary BS forwards the listened primary signals inorder to increase the primary spectrum accessing oppor-tunities, which leads to completely different problem andstrategy design from ours.

• In narrowband single-user case, where the MBS servesa single MUE and each PBS serves a single PUE, wefirst find the explicit expressions of the optimal fICICprecoders, which maximize the signal-to-interference-plus-noise ratio (SINR) of the PUE in each pico cell.We then analyze the asymptotical performance of thefICIC in ICI-dominated scenario. The results showthat under perfect self-interference cancellation for FD,the fICIC can thoroughly eliminate weak ICI, whilewhen the ICI is very strong or the residual self-interference is very large, the fICIC will reduce to the HDscheme.

• In narrowband multi-user case, where the MBS servesmultiple MUEs and each PBS serves multiple PUEs,we propose a low-complexity algorithm to optimize thefICIC scheme, aimed at maximizing the downlink sumrate of each pico cell under the fairness constraint overthe PUEs. Simulations validate the analytical results,and demonstrate a significant performance gain of thefICIC over the HD scheme as well as evident benefitsof combining the fICIC with existing eICIC and CoMPtechniques.

• We generalize the narrowband fICIC to orthogonalfrequency-division multiplexing (OFDM) systems, wherethe optimization problem for the fICIC precoder designaimed at maximizing sum rate over multiple subcarriersis obtained, which can be solved with a gradient basedmethod. Simulations shows the advantages of the fICICon mitigating wideband ICI.

Notations: (·)T , (·)∗, and (·)H denote transpose, complexconjugate, and conjugate transpose, respectively. X � 0 repre-

sents that matrix X is positive semi-definite, and X12 denotes

hermitian square root of X. tr(·) denotes matrix trace, rank(·)denotes matrix rank, E{·} denotes expectation operator, vec(·)denotes vectorization operator, ⊗ denotes Kronecker product,� denotes convolution product, and ‖ · ‖ denotes Euclidiannorm. diag(X) denotes the diagonal matrix with the same diag-onal elements as X. IN , 0N and 0N denote N × N identity and

TABLE ILIST OF MAJOR SYMBOLS

zero matrices, and N × 1 zero vector, respectively. The majorsymbols used in the paper are listed in Table I.

II. SYSTEM MODEL

We first consider the downlink transmission of a narrowbandtime division duplex (TDD) HetNet, and then generalize themodel to wideband systems. Suppose that the HetNet consistsof one MBS and B PBSs, where the MBS serves KM single-antenna MUEs and each PBS serves K P single-antenna PUEs.Assume that the MUEs experience negligible interference fromPBSs due to the coverage range expansion (CRE) of pico cells,and the pico cells are geographically separated so that each PUEreceives much weaker interference from interfering PBSs com-pared to the interference generated by the MBS, which is treatedas noise in the paper. Therefore, we focus on the suppressionof the cross-tier ICI generated by the MBS to PUEs, which iscommonly recognized as a bottleneck to improve the spectralefficiency in real-world HetNets [1].

We consider applying FD technique at each PBS in thedownlink transmission, with which the PBS can send thedesired signals and forward the listened cross-tier ICI simul-taneously. The structure of the FD PBS transceiver is illustratedin Fig. 1(a), where uplink and downlink data flows are indicatedby dashed and solid lines, respectively. The FD PBS is com-prised of traditional baseband (BB) and radio frequency (RF)modules in HD transceiver, taking charge of the transmissionand reception of desired signals of PUEs, as well as additionalFD modules, dedicated for self-interference cancellation andICI suppression. The antennas at the PBS can be divided intotwo parts as shown in Fig. 1(a), where the “FD receive anten-nas” are only used to receive the ICI from the MBS in thedownlink, while the “transmit antennas” and the “HD receiveantennas” share the same antennas in a TDD manner, whichare used to send and receive the signals of PUEs in downlinkand uplink, respectively. Therefore, the uplink and downlinkchannel reciprocity holds between the PBS and the PUE.


Fig. 1. (a) System model of the TDD HetNet, where the transceiver structure of the FD PBS is illustrated in the left part. (b) Illustration of the considered HetNetlayout, where the MBS serves two MUEs and the reference PBS serves two PUEs.

Since the proposed fICIC scheme will not affect the perfor-mance of MUEs and other-cell PUEs, in the sequel we onlyconsider a reference PBS and focus on the performance ofthe PUEs served by the reference PBS. The resulting interfer-ence environment is demonstrated in Fig. 1(b). Suppose thatthe MBS has M transmit antennas, M ≥ KM , and the FD PBShas Nt transmit antennas, Nt HD receive antennas, and Nr

FD receive antennas, Nt ≥ K P . Let hMk ∈ CM×1 and hPk ∈CNt ×1 denote the channels from the MBS and the PBS to thek-th PUE (denoted by PUEk), HM P ∈ CM×Nr denote the chan-nel from the MBS to the PBS, and HP P ∈ CNt ×Nr denote theself-interference channel of the FD PBS.

A. Signals of the FD PBS

In the downlink the FD PBS can transmit and receive sig-nals simultaneously. We use the index t to denote time instant.To reflect the impact of hardware impairments of transmitterchains on self-interference cancellation, we can express thereceive signal at the PBS before self-interference cancellationbased on [8], [9] as

yp[t] = HHM P WM sM [t] + HH

P P

(xp[t] + zx [t]

)+ np[t], (1)

where WM ∈ CM×KM is the precoding matrix at the MBSfor sending the signals sM ∼ CN(0KM , IKM ) to the MUEs,the term HH

P P

(xp[t] + zx [t]

)is the self-interference, xp ∈

CNt ×1 is the transmit signal vector of the FD PBS, zx ∼CN(0Nt , μx diag(�x )) is the transmitter distortion with �x

denoting the covariance matrix of xp, μx 1 is a scalingconstant, which reflects the combined effects of additive power-amplifier noise, non-linearities in digital-to-analog converterand power amplifier, I/Q imbalance, and oscillator phase noise,and np ∼ CN(0Nr , σ

2n INr ) is the additive white Gaussian noise

(AWGN), which takes into account both thermal noises and theICI from other PBSs.

Further considering the hardware impairments of receiverchains based on [8], [9], the distorted receive signal canexpressed as

y′p[t] = yp[t] + zy[t], (2)

where zy ∼ CN(0Nr , μydiag(�y)) is the additive distortioncaused by adaptive gain control noise, non-linearities in analog-to-digital converter and gain control, I/Q imbalance, and oscil-lator phase noise in receiver chains, μy 1 is a scalingconstant, and �y is the covariance matrix of the undistortedreceive signal yp, which can be obtained from (1) as

�y = EsM ,xp,zx ,np {yp[t]yHp [t]} = HH

M P WM WHM HM P

+ HHP P (�x + μx diag(�x ))HP P + σ 2

n INr . (3)

Since the transmit signal xp is known at the FD PBS, theself-interference HH

P P xp in (1) can be cancelled if the self-interference channel HP P is estimated. During a dedicatedtraining phase, the orthogonal pilot signals

√Ptr C are trans-

mitted for self-interference channel estimation, where Ptr isthe transmit power of pilot signals, and C = [c1, . . . , cNt ] ∈CNt ×Nt is unitary with CCH = INt . Then, similar to (1) and(2), we can obtain the receive pilot signals with hardwareimpairments of both transmitter and receiver chains as

Y[t] = Y[t] + Zy[t] � HHP P

(√Ptr C + Zx [t]

)+ N[t] + Zy[t], (4)

where Y = [y1, . . . , yNt ] ∈ CNr ×Nt denotes the undis-torted receive signal, Zy = [zy,1, . . . , zy,Nt ] ∈ CNr ×Nt

is the additive distortion of receive signals withzy,i ∼ CN(0Nr , μydiag(�y,i )), �y,i is the covariancematrix of yi , Zx = [zx,1, . . . , zx,Nt ] ∈ CNt ×Nt is the trans-mitter distortion with zx,i ∼ CN(0Nt , μx Ptr diag(ci cH

i )),and N ∈ CNr ×Nt is the AWGN whose columns follow thedistribution CN(0Nr , σ

2n INr ). It can be obtained from (4) that

�y,i = Ptr HHP P

(ci cH

i + μx diag(ci cHi ))

HP P + σ 2n INr .

With least-squares channel estimator, we can estimate theself-interference channel as

HP P = 1√Ptr

CYH [t] = HP P + 1√Ptr

(CZH

x [t]HP P

+ CNH [t] + CZHy [t]

)� HP P + EP P [t], (5)


where EP P = [eP P,1, . . . , eP P,Nt ]H ∈ CNt ×Nr denotes the

channel estimation errors.Denoting ci = [ci1, . . . , ci Nt ]

T , we can express eP P,i as

eP P,i = 1√Ptr

⎛⎝HH

P P

Nt∑j=1

ci j zx, j + Nci +Nt∑

j=1

ci j zy, j

⎞⎠ , (6)

from which we can obtain that eP P,i follows the distributionCN(0Nr , �e,i ) with

�e,i = HHP P

Nt∑j=1

|ci j |2μx diag(c j cHj )HP P + (1 + μy)σ

2n

PtrINr +

Nt∑j=1

|ci j |2μydiag(

HHP P

(c j cH

j + μx diag(c j cHj ))

HP P

).

(7)

With HP P , the receive signal at the PBS after self-interference cancellation is

yp[t] = y′p[t] − HH

P P [t]xp[t] � HHM P sM [t] − EH

P P [t]xp[t]

+ HHP P zx [t] + np[t] + zy[t], (8)

where HM P � WHM HM P is the equivalent channel from the

MBS to the FD PBS.The FD PBS then transmits the desired signals of the PUEs

together with the self-interference cancelled receive signals yp.The combined transmit signal of the PBS can be expressed as

xp[t] = W f yp[t − τ ] +K P∑k=1

wd,ksp,k[t], (9)

where τ is the processing delay introduced by the FD mod-ules, W f ∈ CNt ×Nr is the precoding matrix for the forwardedsignals, wd,k ∈ CNt ×1 is the precoding vector for the desiredsignal, sp,k , of PUEk , and sp,k ∼ CN(0, 1).

In (9), the receive signal yp by Nr additional FD receiveantennas of the PBS are forwarded via W f . The forwarded sig-nal will occupy a part of transmit power of the PBS, whichleads to the reduction of the transmit power for desired sig-nals. As will be clear later, however, the forwarded signal can beused to mitigate the cross-tier ICI at PUEs efficiently with theoptimized W f , and hence result in the improvement of PUEs’data rate.

With (8) and (9) we can calculate the transmit power of theFD PBS as

Pout = tr(�x ) = tr(EsM ,sp,k ,np,zx ,zy ,EP P ,HP P {xp[t]xH

p [t]})

,

(10)

where the expectations are taken over data sM and sp,k , noisesnp, transmitter and receiver distortions zx and zy , channelestimation errors EP P , and self-interference channel HP P ,respectively.1

1In order to highlight the benefits of the fICIC scheme via forwarding thelistened ICI, we restrict ourselves to the case where the precoders W f and{wd,k } are designed only for transmitting the listened ICI and desired signalsbut not for spatial-domain self-interference cancellation (i.e., design W f and{wd,k } to pre-null self-interference). Towards this end, we take the expectationover HP P in (10), such that the precoders are independent of the instantaneousself-interference channel HP P .

Assume that HP P follows Rayleigh distribution, i.e.,vec(HP P ) ∼ CN(0Nr Nt , αP P INr Nt ) with αP P denoting theaverage channel gain, which is reasonable because the trans-mit antennas and the FD receive antennas can be well isolatedin the considered scenario as will be detailed in Section VI-C.Then, we show in Appendix A that the transmit power Pout canbe expressed as

Pout ≈ tr(

W f HHM P HM P WH

f

)+

K P∑k=1

‖wd,k‖2

+(σ 2

n + σ 2e Pout

)tr(W f WH

f ), (11)

where the approximation follows from μx 1 and μy 1

as in [8], and σ 2e = σ 2

nPtr

+ 2αP P (μx + μy) reflects the residual

self-interference, in which the term σ 2n

Ptrcomes from imperfect

channel estimation and the term 2αP P (μx + μy) comes fromhardware impairments.

From the equation with respect to Pout given in (11), we canobtain the transmit power of the FD PBS as

Pout = (12)

tr(

HM P WHf W f HH

M P

)+ σ 2

n tr(

WHf W f

)+∑K P

k=1 ‖wd,k‖2

1 − σ 2e tr

(WH

f W f

) ,

which shows that the power of the precoding matrix W f for ICIforwarding must be limited by

tr(

WHf W f

)<

1

σ 2e

. (13)

Otherwise, amplifier self-oscillations will occur at the FDPBS [7].

Let P0 denote the maximal transmit power of the PBS. Thenthe transmit power Pout needs to satisfy Pout ≤ P0, which canbe rewritten based on (12) as

tr(

HM P WHf W f HH

M P

)+ (P0σ

2e + σ 2

n )tr(

WHf W f

)

+K P∑k=1

‖wd,k‖2 ≤ P0. (14)

One can observe that the self-oscillation constraint (13)is implicitly reflected in the maximal power constraint (14).Therefore, in the following only the constraint in (14) isconsidered.

B. Signals of PUEk

The receive signal of PUEk can be expressed as

yk[t] = hHPkxp[t] + hH

MkWM sM [t] + nk[t], (15)

where the impact of hardware impairments at the HD PUE isignored as commonly considered for HD transmission in theliterature, and nk[t] ∼ CN(0, σ 2

n ) is the AWGN at PUEk , whichincludes the received ICI from interfering PBSs and thermalnoises.


By noting the equivalence between time delay and phaseshift in narrowband systems, we have sM [t − τ ] = sM [t]e− jφ ,where φ = 2π fcτ with fc denoting the carrier frequency.2 Thenbased on (8) and (9), we can rewrite (15) as

yk[t] = hHPkwd,ksp,k[t] +

K P∑j=1, j �=k

hHPkwd, j sp, j [t]

︸︷︷︸Intra-cell interference

+ hHPkW f

(−EH

P P [t − τ ]xp[t − τ ] + HHP P zx [t − τ ]+

np[t − τ ] + zy[t − τ ])

︸︷︷︸Forwarded residual self-interference and noises

+(

hHMk + hH

PkW f HHM P e− jφ

)sM [t]︸︷︷︸

Cross-tier ICI

+nk[t], (16)

where hMk � WHM hMk is the equivalent channel from the MBS

to PUEk .To achieve coherent combination between the ICI forwarded

by the PBS and that directly received at the PUEs as shownin (16), the PBS needs to implement sample-by-sample ICIforwarding in time domain to reduce the processing delay asconsidered in [10], [11] for relay systems, where the time-domain FD amplify-and-forward relay was employed to pro-vide co-phasing combining gain at the destination node. Thisis different from frequency domain forwarding for FD relaysystems, e.g., in [7] by the same authors as [10], [11], wherethe processing delay will be in symbol-level because a com-plete symbol needs to be received and demodulated beforeforwarding in the frequency domain.

Similar to the derivations in Appendix A, we can compute thesignal and interference power from (16) by taking the expecta-tion with respect to sM , sp,k , np, zx , zy , EP P , and HP P , andfinally obtain the SINR of PUEk as

SINRk,F D =

|hHPkwd,k |2/

⎛⎝∑

j �=k

‖hHPkwd, j‖2 + ‖hH

Mk + hHPkW f HH

M P e− jφ‖2

+‖hHPkW f ‖2(Poutσ

2e + σ 2

n ) + σ 2n

⎞⎠ , (17)

where Pout is the transmit power of the PBS, which is a functionof the precoders W f and wd,k as given in (12).

III. NARROWBAND SINGLE USER CASE

In this section, we investigate the single-user case, i.e.,KM = K P = 1, to gain insight into the ICI mitigation mech-anism of the fICIC scheme.

In this case, the equivalent channel HM P is a row vectorand hMk is a scalar, which are denoted by hH

M P ∈ C1×Nr and

2One can further incorporate the propagation delay difference experiencedby the forwarded ICI and the direct ICI into τ , which is not consideredhere because the PUEs are close to the PBS leading to negligible additionalpropagation delay.

hMk ∈ C1×1 for clarity, respectively. Moreover, noting that theintra-cell interference does not exist now, the SINR can besimplified as

SINRk,F D =|hH

Pkwd,k |2|h�

Mk+hHPkW f hM P e− jφ |2+‖hH

PkW f ‖2(Poutσ 2e +σ 2

n )+σ 2n

,

(18)

where Pout given in (12) can be rewritten as

Pout =‖W f hM P‖2 + σ 2

n tr(

WHf W f

)+ ‖wd,k‖2

1 − σ 2e tr

(WH

f W f

) . (19)

When the precoder for forwarding the listened ICI, W f , in(18) is selected as zero, the FD PBS reduces to a HD PBS. Itis not hard to obtain the optimal precoder for transmitting thedesired signal in HD case that maximizes the SINR of PUEk as

wd,k =√

P0hPk‖hPk‖ , which is referred to as “the HD scheme” in the

sequel. The maximum SINR achieved by the HD scheme canbe obtained from (18) as

SINR�k,H D = P0‖hPk‖2

|hMk |2 + σ 2n

. (20)

Compared to the HD scheme where the ICI power is |hMk |2as shown in (20), the FD scheme turns the ICI power into|h�

Mk + hHPkW f hM P e− jφ |2 as shown in (18), which can be

reduced by optimizing the precoders W f and wd,k . The opti-mization problem, aimed at maximizing the SINR of PUEk

subject to the maximal transmit power constraint, can be for-mulated as

maxW f ,wd,k

SINRk,F D (21a)

s. t. ‖W f hM P‖2 + (P0σ2e + σ 2

n )tr(

WHf W f

)+ ‖wd,k‖2 ≤ P0, (21b)

where the power constraint (21b) comes from (14).

A. Optimal fICIC Scheme

In this subsection, we strive to find the optimal fICIC schemewith explicit expressions for W f and wd,k . Given that it isdifficult to directly solve problem (21) due to the complexexpression of SINRk,F D in (18), we start with investigatingthe properties of the optimal solutions to problem (21) in thefollowing.

First, by substituting the expression of Pout given in (19) into(18), we can find that SINRk,F D , i.e., the objective function ofproblem (21), is an increasing function of ‖wd,k‖2. Therefore,for any given W f and the direction of wd,k , wd,k

‖wd,k‖ , we can

always improve the SINR by increasing ‖wd,k‖2 until the PBStransmits with its maximum power. This means that the globaloptimal solution to problem (21) is obtained when the powerconstraint in (21b) holds with equality.


Based on this result, we can obtain from (19) that Pout = P0,with which SINRk,F D is simplified as

SINRk,F D =|hH

Pkwd,k |2|h�

Mk + hHPkW f hM P e− jφ |2 + ‖hH

PkW f ‖2(P0σ 2e + σ 2

n ) + σ 2n

.

(22)

Second, because the direction of wd,k affects only the numer-ator of SINRk,F D , we can obtain that the optimal w�

d,k satisfiesw�

d,k

‖w�d,k‖

= hPk

‖hPk‖ . (23)

Third, we can show that the optimal W�f has the following

property.Lemma 1: The optimal W�

f is of rank 1, which can bedecomposed as

W�f = −h�

Mke jφ · β · hPk hHM P , (24)

where β is a positive scalar.

Proof: See Appendix B. �We can see from the expression of the optimal W�

f that withthe fICIC the listened ICI is first enhanced by the maximal ratiocombining with hH

M P , and then forwarded by the maximal ratiotransmission with hPk .

Based on (23) and (24), we can express the SINR with β

and ‖wd,k‖. Further considering that the optimal solution isobtained when the power constraint in (21b) holds with equal-ity, we can replace ‖wd,k‖ with β, and finally convert problem(21) into the following optimization problem

maxβ

f0(β) (25a)

s. t.(‖hM P‖2 + σ 2

I + σ 2n

)|hMk |2‖hPk‖2‖hM P‖2β2 ≤ P0

(25b)

β ≥ 0, (25c)

where f0(β) �(‖hH

Pk‖2(P0 − (‖hM P‖2 + σ 2

I + σ 2n

) |hMk |2‖hPk‖2‖hM P‖2β2

))/((|hMk | − β|hMk |‖hPk‖2‖hM P‖2

)2 + β2

|hMk |2‖hPk‖4‖hM P‖2(σ 2I + σ 2

n ) + σ 2n

), and σ 2

I � P0σ2e

denotes the average power of residual self-interference.Proposition 1: The maximum of the objective function in

(25a), i.e. the maximal SINR of PUEk , can be expressed as

SINR�k,F D = 1

1Bβ� − 1

, (26)

with β� =2

(A+D−

√(A+D)2− AC2

B

)C2 representing the optimal

solution of β, where A = P0‖hPk‖2, B = ‖hPk‖2(‖hM P‖2 +σ 2

I + σ 2n ), C = 2|hMk |‖hPk‖‖hM P‖, and D = |hMk |2 + σ 2

n .

Proof: See Appendix C. �With the optimal β�, we can directly obtain the optimal W�

ffrom (24). To compute the optimal w�

d,k , recalling that con-straint (21b) holds with equality for the optimal solutions, wecan obtain the norm of the optimal w�

d,k as

‖w�d,k‖ =

√P0 − 1

4

(‖hM P‖2 + σ 2I + σ 2

n

)C2β∗2. (27)

Then by substituting (27) into (23), we can obtain w�d,k .

B. Asymptotic Performance Analysis

To understand the behavior of the fICIC scheme, we nextconsider an ICI-dominated scenario, where the noise power σ 2

napproaches to zero.

1) Perfect Self-Interference Cancellation: If the self-interference can be perfectly eliminated, i.e., σ 2

I = 0, fromthe definitions after (26) we have B

.= ‖hPk‖2‖hM P‖2 andD

.= |hMk |2, where.= denotes asymptotic equality. Then based

on Proposition 1, after some manipulations we can obtain theoptimal β� as

β� .= P0‖hPk‖2 + |hMk |2 − ∣∣P0‖hPk‖2 − |hMk |2∣∣

2‖hM P‖2|hMk |2‖hPk‖2(28)

= min(P0‖hPk‖2, |hMk |2)‖hM P‖2|hMk |2‖hPk‖2

� η

‖hM P‖2|hMk |2‖hPk‖2.

By substituting (28) into (25a), the maximal SINR of PUEk

becomes

SINR�k,F D

.= P0|hMk |2‖hPk‖2 − η2

(η − |hMk |2)2 +(

η2

‖hM P‖2 + |hMk |2)

σ 2n

, (29)

where η = min(P0‖hPk‖2, |hMk |2) as defined in (28), whichdepends on the strengths of the desired signal, P0‖hPk |2, andthe ICI, |hMk |2. In the following two cases are discussed.

• Case 1: |hMk |2 < P0‖hPk‖2

This is a case where the ICI is weaker than the desiredsignal. Then, we have η = |hMk |2 and the maximal SINRis

SINR�k,F D

.= P0‖hPk‖2 − |hMk |2( |hMk |2‖hM P‖2 + 1

)σ 2

n

. (30)

It implies that the FD PBS can thoroughly eliminate theICI generated by the MBS by properly designing theforwarding and transmitting precoders.When compared with the HD scheme, the performancegain of the fICIC can be obtained as

SINR�k,F D

SINR�k,H D

.=1 − |hMk |2

P0‖hPk‖2(1

‖hM P‖2 + 1|hMk |2

)σ 2

n

. (31)

• Case 2: |hMk |2 ≥ P0‖hPk‖2

In this case where the ICI is stronger, we have η =P0‖hPk‖2 and the maximal SINR is

SINR�

k,F D.= P0‖hPk‖2

|hMk |2 − P0‖hPk‖2 +P2

0 ‖hPk‖4

‖hM P ‖2 +|hMk |2|hMk |2−P0‖hPk‖2 σ 2

n

.

(32)

For very strong interference, i.e., |hMk |2 � P0‖hPk‖2,SINR

∗k,F D can be approximated as

SINR∗k,F D ≈ P0‖hPk‖2

|hMk |2.= SINR∗

k,H D. (33)

From the above analysis, we can obtain the followingobservations.


• Impact of hM P : It is shown from (30) and (32) that theSINR of PUEk increases with the channel gain betweenMBS and PBS ‖hM P‖. This is because given the powerof the PBS allocated for forwarding the listened ICI,denoted as

P f wout � ‖W f hM P‖2 + (σ 2

I + σ 2n )tr

(WH

f W f

), (34)

a large ‖hM P‖ will reduce the power used for forwardingthe residual self-interference and noises and thus improvethe efficiency of power usage. However, when the ICIpower |hMk |2 is very large, the impact of ‖hM P‖ can beneglected as shown in (33).

• Impact of hPk and hMk : As shown by (20), (30)and (33), increasing the desired signal P0‖hPk‖2 canimprove the performance of both the FD scheme and theHD scheme.3 However, the performance improvement forthe FD scheme is more significant because (31) showsthat the performance gain of FD over HD increases withP0‖hPk‖2. It can also be seen that the performance gainof FD over HD decreases with the ICI power |hMk |2, untilvanishes for very large |hMk |2 as shown by (33). It shouldbe pointed out that the extreme case with very strong ICIin (33) rarely happens in practice even when the maxi-mum CRE offset of 9 dB in LTE systems is considered[12]. As will be verified in Fig. 4, the fICIC still exhibitsevident performance gain over the HD scheme when theaverage power of the ICI is 10.4 dB stronger than thedesired signal.

2) Large Residual Self-Interference: When the self-interference cancellation for FD is imperfect and the residualself-interference, σ 2

I , is large, the parameter B is large and

the term AC2

B is small. Then by using the first-order taylorexpansion,

√c − z

.= √c − 1

2√

cz for small z, we can obtain

β∗ .= A(A+D)B when AC2

B approaches to zero, with which wecan obtain from (26) the maximal SINR of PUEk as

SINR∗k,F D

.= P0‖hPk‖2

|hMk |2 + σ 2n

= SINR∗k,H D. (35)

Moreover, we can compute the power of the PBS allocated forforwarding the listened ICI from (34) as

P f wout

.= P20 ‖hM P‖2|hMk |2‖hPk‖2

(|hMk |2 + P0‖hPk‖2 + σ 2n )2(‖hM P‖2 + σ 2

I + σ 2n )

.

(36)

It can be seen from (36) that the transmit power of the PBSallocated for forwarding ICI decreases with the growth of resid-ual self-interference σ 2

I . When σ 2I is very large, the PBS will

use all power to transmit desired signals, and therefore thefICIC will reduce to the HD scheme.

3Herein, we consider that the increase of the desired signal’s strength comesfrom reducing the distance between the PUE and its serving PBS, but not fromincreasing the power of the PBS. As a result, the interference from surroundingPBSs to the PUE will be even weaker, making it reasonable to focus on thedominant cross-tier ICI as we considered.

IV. NARROWBAND MULTI-USER CASE

In this section, we consider the general multi-user case,where the MBS serves KM MUEs and the PBS serves K P PUEswith KM ≥ 1 and K P ≥ 1.

We optimize the fICIC scheme, aimed at maximizing the sumrate of K P PUEs served by the reference PBS while guaran-teeing the fairness among the PUEs, subject to the maximaltransmit power constraint of the PBS. The problem can beformulated as

maxW f ,{wd,k }

Rsum (37a)

s. t. log(1 + SINRk,F D) = αk Rsum, k = 1, . . . , K P (37b)

tr(

HM P WHf W f HH

M P

)+ (σ 2

I + σ 2n )tr

(WH

f W f

)

+K P∑k=1

‖wd,k‖2 ≤ P0, (37c)

where Rsum is the sum rate of all PUEs, the constraints in (37b)ensure the data rate proportion among the PUEs with prede-fined fairness factors {αk} as considered in [13], αk ≥ 0, and∑K P

k=1 αk = 1.The expression of SINRk,F D in multi-user case is given in

(17), which is a function of the transmit power Pout as shownin (12) and thus is very complicated. To simplify SINRk,F D , wecan prove that the optimal solution to problem (37) is obtainedwhen the power constraint (37c) holds with equality.4 Based onthis result, we have Pout = P0 in (17).

Problem (37) is to find the maximal achievable Rsum ,denoted by R∗

sum , ensuring all constraints satisfied, whichcan be solved by bisection methods [14]. Specifically, for agiven Rsum in an iteration, denoted by R0

sum , if the followingoptimization problem

Find W f , {wd,k} (38a)

s. t. (37c)

SINRk,F D = 2αk R0sum − 1, k = 1, . . . , K P (38b)

is feasible, then it follows that R0sum is an achievable sum rate

of all K P PUEs, i.e., R0sum ≤ R∗

sum , otherwise, R0sum > R∗

sum .This condition can be used in bisection algorithms to find R∗

sum .Now the remaining issue is to find efficient approaches to

evaluate the feasibility of problem (38). In the following, weshow that the feasibility problem can be solved by investigatingthe optimization problem below

minW f ,{wd,k }

tr(

HM P WHf W f HH

M P

)+ (σ 2

I + σ 2n )·

tr(

WHf W f

)+

K P∑k=1

‖wd,k‖2 (39a)

s. t. SINRk,F D ≥ γk, k = 1, . . . , K P , (39b)

4Otherwise, suppose that the power constraint holds with inequality withthe optimal precoders W∗

f and {w∗d,k }, then given W∗

f , we can always find

new precoders {w′�d,k }, defined as w′�

d,k = cw∗d,k for k = 1, . . . , K P with c > 1,

which can further improve the SINR of all PUEs until the constraint (37c) holdswith equality.


where the objective function is the left-hand side of constraint(37c), and γk � 2αk R0

sum − 1.To see this, we note that the optimal solution to problem (39)

is obtained when all SINR constraints in (39b) hold with equal-ity, otherwise, the value of the objective function can be alwaysfurther reduced by properly decreasing ‖wd,k‖2. It suggests thatif the minimum of the objective function (39a) is smaller thanP0, then constraints (37c) and (38b) in problem (38) are all sat-isfied. This means that problem (38) is feasible and R0

sum is anachievable sum rate, otherwise, problem (38) will be infeasible.In what follows, we solve problem (39).

By defining w f = vec(WHf ), we can rewrite problem (39) as

minw f ,{wd,k }

‖ (INt ⊗ HM P)

w f ‖2+(σ 2I +σ 2

n )‖w f ‖2+K P∑k=1

‖wd,k‖2

(40a)

s. t.|hH

Pkwd,k |2�k

≥ γk, ∀k, . (40b)

where �k�∑

j �=k‖hHPkwd, j‖2+‖hMk+e− jφ(hT

Pk⊗HM P )w f ‖2

+‖ (hTPk⊗INr

)w f ‖2(σ 2

I + σ 2n ) + σ 2

n .Problem (40) is non-convex because the constraints in (40b)

are non-convex. A common method to solve such a problemis to convert the non-convex SINR constraints into convexsecond-order cone constraints [15]. Specifically, since addingany phase rotation to wd,k will not affect the SINR of all PUEs,we can assume that hH

Pkwd,k is real-valued, which does notaffect the global optima of problem (40). Then we can convertthe non-convex problem (40) into the following second-ordercone constrained problem

minw f ,wd

‖ (INt ⊗HM P)

w f ‖2+(σ 2I +σ 2

n )‖w f ‖2+‖wd‖2 (41a)

s. t.

√1 + 1

γkhH

Pkwd,k

≥

∥∥∥∥∥∥∥∥∥

⎡⎢⎢⎢⎣

(INr ⊗ hH

Pk

)wd

e− jφ(hT

Pk ⊗ HM P)

w f√σ 2

I + σ 2n

(hT

Pk ⊗ INr

)w f

σn

⎤⎥⎥⎥⎦+

⎡⎢⎢⎣

0Nr

hMk

0Nr

0

⎤⎥⎥⎦∥∥∥∥∥∥∥∥∥

,∀k, (41b)

where wd = vec([wd,1, . . . , wd,K P ]

).

The resultant problem (41) is convex since both the objec-tive function and constraints are convex, which can be solvedby standard convex optimization algorithms [14]. However, thecomputational complexity of the standard algorithms is still toohigh and prohibits the practical use of the fICIC scheme, espe-cially when the numbers of MUEs, PUEs, and the transmit andreceive antennas at the PBS are large. In the following we striveto propose a low-complexity algorithm to solve problem (40),with which the optimal precoders are obtained with explicitexpressions. We begin with the discussion regarding the strongduality of problem (40).

Recalling that we have shown the equivalence between prob-lem (40) and problem (41), and also known that strong dualityholds for problem (41) because it is a convex problem. Further,along the lines of [15, App.A], we can show the equivalence

between the Lagrangian functions of problem (41) and problem(40). Therefore, strong duality holds for the non-convex opti-mization problem (40). This means that we can solve problem(40) with the Lagrange dual method.

The dual problem to problem (40) can be expressed as

maxλk

minw f ,{wd,k }

J (λk, w f , wd,k) (42a)

s.t. λk ≥ 0, ∀k, (42b)

where λk is the lagrangian multiplier, and J (λk, w f , wd,k) isthe Lagrangian function of problem (40) with the expression

J (λk, w f , wd,k) =∥∥(INt ⊗ HM P)

w f∥∥2 + (σ 2

I + σ 2n )‖w f ‖2

+K P∑k=1

λk

(‖hMk + e− jφ

(hT

Pk ⊗ HM P

)w f ‖2

+‖(

hTPk ⊗ INr

)w f ‖2(σ 2

I + σ 2n ) + σ 2

n

)

+K P∑k=1

wHd,k

⎛⎝INt +

∑j �=k

λ j hP j hHP j − λk

γkhPkhH

Pk

⎞⎠wd,k . (43)

A. Optimal Solution to λk

It can be seen from (43) that minw f ,{wd,k }

J (λk, w f , wd,k) →−∞ except when INt +∑

j �=k λ j hP j hHP j − λk

γ0hPkhH

Pk � 0Nt .Hence, the dual problem (42) is equivalent to

minw f

maxλk

J (λk, w f , 0Nt ) (44a)

s. t. INt +∑j �=k

λ j hP j hHP j − λk

γkhPkhH

Pk � 0Nt ,∀k (44b)

λk ≥ 0,∀k, (44c)

where the objective function (44a) comes from the fact thatJ (λk, w f , wd,k) given in (43) is minimized when wd,k = 0Nt .

It is difficult to directly find the optimal λk from problem (44)due to the complicated semi-definite positive constraints (44b).To solve the problem, we simplify the constraints as follows.

Proposition 2: The semi-definite positive constraints (44b)can be equivalently expressed as

λkhHPk

⎛⎝INt +

∑j �=k

λ j hP j hHP j

⎞⎠−1

hPk ≤ γk,∀k. (45)

Proof: See Appendix D. �Note that for any given w f , the objective function of prob-

lem (44) is an increasing function of λk . Then we can showthat the optimal λk maximizing J (λk, w f , 0Nt ) is obtainedwhen the constraints in (45) hold with equality (otherwise, onecan always increase λk to improve the value of the objectivefunction). It suggests that the optimal λ∗

k should satisfy

λ∗k = γk

hHPk

(INt +∑

j �=k λ∗j hP j hH

P j

)−1hPk

. (46)


(46) provides a fixed-point iterative algorithm to find theoptimal λ�

k , which can be expressed as

λ∗(n+1)k = γk

hHPk

(INt +∑

j �=k λ�(n)j hP j hH

P j

)−1hPk

, (47)

where the superscript (n) denotes the n-th iteration.The convergence of the fixed-point iterative algorithm can

be proved based on the standard function theory [16], whichshows that the algorithm given in (47) will converge to a uniqueoptimal solution from any initial value {λ�(0)

k }.

B. Optimal Solution to w f

After obtaining the optimal λ�k for k = 1, . . . , K P , we

next find the optimal w�f based on the Karush-Kuhn-Tucker

(KKT) condition [14] associated with the Lagrangian functionJ (λ�

k, w f , wd,k). We can obtain that the optimal w�f satisfies

K P∑k=1

e jφλ∗k

(hT

Pk ⊗ HM P

)HhMk +

(INt ⊗ HH

M P HM P

)w∗

f

+ (σ 2I + σ 2

n )w∗f +

K P∑k=1

λ∗k

(((hPkhH

Pk)T ⊗ HH

M P HM P

)w∗

f

+(σ 2I + σ 2

n )((hPkhH

Pk)T ⊗ INr

)w∗

f

)= 0Nt Nr . (48)

By applying the properties of Kronecker product [17] andafter some manipulations, we can obtain the optimal w�

f as

w�f = −e jφ

⎛⎝( K P∑

k=1

λ∗k(hPkhH

Pk)T +INt

)−1( K P∑k=1

λ∗k hMkhH

Pk

)T

⊗(

HHM P HM P + (σ 2

I + σ 2n )INr

)−1HH

M P

⎞⎠ vec(INt Nr ). (49)

Then we can recover the optimal W∗f from w∗

f as

W�f = − e jφ

( K P∑k=1

λ�khPkhH

Pk + INt

)−1

· (50)

( K P∑k=1

λ�khPk hH

Mk

)HM P

(HH

M P HM P + (σ 2I + σ 2

n )INr

)−1.

C. Optimal Solution to wd,k

With the optimal λ�k and W�

f , the optimal w�d,k of problem

(40) can be solved as follows.According to the KKT condition, the optimal w�

d,k satisfies

∂ J (λk�, w�

f , wd,k)

∂wd,k(51)

= 2

⎛⎝INt +

∑j �=k

λ�j hP, j hH

P, j − λ�k

γkhPkhH

Pk

⎞⎠wd,k = 0Nt ,

TABLE IILOW-COMPLEXITY ALGORITHM FOR MULTI-USER FICIC

from which we have

w�d,k =λ�

khHPkw�

d,k

γk

⎛⎝INt +

∑j �=k

λ�j hP, j hH

P, j

⎞⎠−1

hPk

�√

p�kw�

d,k, (52)

where w�d,k =

(INt +∑

j �=k λ�j hP, j hH

P, j

)−1hPk , and p�

k is a

scalar controlling the power allocated to PUEk for transmittingdesired signals.

To find the optimal {p�k} in (52), recalling that the optimal

solution to problem (40) is obtained when all SINR constraintsin (40b) hold with equality if problem (40) is feasible for given{γk}, then we can obtain the following equations with respectto p�

k

p�k |hH

Pkw�d,k |2∑

j �=k p�j‖hH

Pkw�t, j‖2 + ζk(W�

f )= γk, k = 1, . . . , K P ,

(53)

where ζk(W�f ) = ‖hH

Mk + hHPkW�

f HHM P e− jφ‖2 + |hH

PkW�f |2

(σ 2I + σ 2

n ) + σ 2n .

From the equations in (53), we can solve the optimal p� �[p�

1, . . . , p�K P

]T as

p� = M−1ζ , (54)

where ζ = [ζ1(W�f ), . . . , ζK P (W�

f )]T , and M ∈ CK P×K P is

defined as

[M]k j ={

1γk

|hHPkw�

d,k |2, k = j,

−|hHPkw�

t, j |2, k �= j.(55)

Finally, we summarize the proposed low-complexity algo-rithm to solve the original optimization problem (37) in generalmultiuser case in Table II.


V. GENERALIZATION TO WIDEBAND SYSTEMS

In previous sections the fICIC is optimized in narrowbandsystems, where all the channels are flat fading and thereforeonly single-tap forwarding precoder is designed. When con-sidering wideband systems, frequency-selective fading chan-nels should be taken into account and hence a multi-tapfinite impulse response (FIR) forwarding precoder needs to bedesigned, which makes the design of the fICIC more involved.

In this section, we generalize the fICIC to OFDM systems.For the sake of notational simplicity, we consider single-usercase, i.e., KM = 1 and K P = 1, which can be straightforwardlyextended to multiuser orthogonal frequency division multi-accessing (OFDMA) systems due to the orthogonality betweensubcarriers. Following the previous narrowband-system defini-tions, let hMk(t) and hPk(t) denote the time-domain channelsfrom the MBS and the PBS to PUEk , HM P (t) denote the time-domain channel from the MBS to the PBS, and HP P (t) denotethe time-domain self-interference channel of the FD PBS. Thecorresponding frequency-domain channels on the n-th subcar-rier of the four channels are denoted by gMkn , gPkn , GM Pn , andGP Pn , respectively.

Consider that the PBS employs a FIR precoder W f (t) =∑L−1l=0 W f lδ(t − lTs) to forward the listened ICI, where L is

the order of the FIR precoder and Ts is the sampling interval.The selection of L needs to ensure that the delay spread of theequivalent channel for the forwarded ICI, hPk(t) � W f (t) �HM P (t) � δ(t − τ), does not exceed the cyclic prefix (CP) ofthe OFDM system in order to maintain orthogonality betweensubcarriers. Let W f n denote the frequency response of W f (t)on the n-th subcarrier.

Then, following the same derivations in Section II fornarrowband systems, we can obtain the transmit power of thePBS on the n-th subcarrier as

Pout,n =tr(

W f n gM Pn gHM PnWH

f n

)+ σ 2

n tr(

W f nWHf n

)+ ‖wdn‖2

1 − σ 2e tr

(W f nWH

f n

) ,

(56)

where gM Pn = GHM PnwMn is the equivalent frequency-domain

channel from the MBS to the PBS, wMn is the precoder at theMBS for the MUE on the n-th subcarrier, wdn is the precoder

at the PBS for the desired signal of PUEk , and σ 2e = σ 2

nPtr

+2αP P (μx + μy) is the same as defined in (11). With (56), thetotal transmit power constraint for the PBS can be expressed as

Pout =N∑

n=1

Pout,n ≤ P0. (57)

Compared to the power constraint in (12) for narrowbandsystem, which can be converted into a convex constraint for theprecoders as shown in (14), the constraint in (57) for widebandsystem is much more complicated and is non-convex.

Similar to (17), we can obtain the SINR of PUEk on the n-thsubcarrier as

SINRkn,F D = |gHPknwdn|2/

(‖g�

Mkn+gHPknW f n gM Pne− jdn‖2

+‖gHPknW f n‖2(Pout,nσ 2

e + σ 2n ) + σ 2

n

),

(58)

where gMkn � wHMngMkn is the equivalent channel from the

MBS to the PUE on the n-th subcarrier, and Pout,n is given in(56), which is a function of the precoders W f n and wdn .

Then, the wideband fICIC precoder optimization problem,aimed at maximizing the sum rate of PUEk over all subcarrier,can be formulated as

maxW f (t),{wdn}

N∑n=1

log(1 + SINRkn,F D) (59a)

s.t.[[W f 1]i j , . . . , [W f N ]i j

]T = F[[W f 1]i j , . . . , [W f L ]i j

]T

(59b)N∑

n=1

Pout,n ≤ P0, Pout,n ≥ 0, ∀n, (59c)

where constraint (59b) restricts that the N frequency-domainprecoders {W f n} are generated from the L-tap time-domainFIR precoder W f (t), and F ∈ CN×L is the matrix containingthe first L columns of the N × N fast fourier transformationmatrix.

Problem (59) is non-convex, whose global optimal solutionis difficult to find. We can obtain a local optimal solutionto the problem by using a gradient-based solution (specifi-cally using the function fmincon of the optimization toolboxof MATLAB). Note that the direction of wdn only affectsthe power of desired signal in the numerator of SINRn,F D .Therefore, the direction of the optimal wdn can be obtained as

w�dn

‖w�dn‖ = gPkn

‖gPkn‖ , with which the number of variables in problem

(59) is reduced.

VI. PRACTICAL ISSUES

By now, we have introduced the concept of fICIC and opti-mized the associated precoders. In this section, we discuss somepractical issues regarding the application of the fICIC.

A. Channel Acquisition

To apply the fICIC, the PBS needs to have the channels fromthe MBS to both the PBS and PUEs, i.e., HM P and hMk , aswell as the channel from the PBS to PUEs, hPk , ∀k. First, thechannel HM P can be directly estimated at the PBS by using theFD receive antennas to receive the downlink training signalssent from the MBS. Second, noting that in TDD systems chan-nel reciprocity holds between the HD transceiver at the PBSand each PUE, the channel hPk can be estimated at the PBSby using the HD receive antennas to receive the uplink train-ing signals sent by PUEk . Finally, the channel hMk can be firstestimated by PUEk and then fed back to the PBS, where digi-tal or analog feedback schemes can be employed [18]. We willevaluate the performance of the fICIC under imperfect channelestimation and feedback in next section.


Fig. 2. Network layout for simulations, where two PBSs are deployed in themacro cell covered by the MBS.

B. fICIC v.s. the HD Scheme

As we analyzed in Section III-B, with the fICIC, the FDPBS can adaptively switch between FD mode and HD mode byoptimizing the precoders for transmitting desired signals andforwarding listened interference. For instance, when the ICI isvery strong or the residual self-interference is very large, thefICIC will reduce to the HD scheme, as shown by (33) and (35).Therefore, with perfect channel information at the PBS, the pro-posed fICIC will always outperform the HD scheme given thesame number of uplink and downlink RF chains and BB mod-ules. Yet, to support the FD technique extra passive antennas(though cheap) and FD modules are required. When imperfectchannels are considered at the PBS, the performance of thefICIC will decrease. Nevertheless, simulations in next sectionshow that significant performance gain can be still achieved bythe fICIC over the HD scheme with imperfect channels.

C. Self-Interference Cancellation

Effective self-interference cancellation is crucial for thefICIC, where isolation of the transmit and FD receive antennasis an important approach [6]. Considering the transceiver struc-ture of the FD PBS given in Fig. 1(a), where the FD receiveantennas are only active in the downlink to receive the ICI fromthe MBS while not receiving the uplink signals from the PUEs,the FD receive antennas can be mounted far away from thetransmit antennas, e.g., installing the FD receive antennas out-side a building and transmit antennas inside, respectively. Inthis manner, the self-interference can be largely suppressed ingeneral.

D. Joint Application With Existing ICIC Schemes

Existing ICIC schemes including eICIC and CoMP-CB oper-ate at the MBS, while the fICIC can operate at each PBS indi-vidually. Therefore, the fICIC can be directly applied togetherwith existing ICIC schemes. As will be shown in next section, ajoint application of the fICIC and existing schemes can achieveevident performance improvement.

VII. SIMULATION RESULTS

In this section we verify our analytical results and evaluatethe performance of the proposed fICIC scheme via simulations.

A. Simulation Setups

For all simulations, unless otherwise specified, the parame-ters in this subsection are used. The considered HetNet layout

Fig. 3. Average sum rate versus SNRedge, where perfect self-interferencecancellation and perfect channels are considered, KM = 1, K P = 1, andNr = 1.

is shown in Fig. 2, where the MBS is located at the centerof the macro cell, PBS1 and PBS2 are respectively located at(dP1, 0) and (dP2, 0), the MBS serves two MUEs located at(dM1, 0) and (dM2, 0), and PBSk serves two PUEs locatedat (dPk, r) and (dPk,−r), respectively. We set the radiusof the macro cell RM as 500 m, dP1 = 60 m, dP2 = 180 m,r = 40 m, dM1 = 120 m, and dM2 = 240 m. Since we haveshown in the analytical results that the performance of thefICIC depends on the strength of the ICI, the positions of thetwo PBSs, dP1 and dP2, are selected to reflect the cases ofstrong and weak ICI, respectively. We will also evaluate theperformance of the fICIC with random PBS placements later.

The MBS transmits with M = 4 antennas and the powerof PM = 46 dBm, each PBS has Nr = 2 FD receive antennasand transmits with Nt = 2 antennas and the maximal power ofP0 = 30 dBm, and each UE (including MUE and PUE) has onereceive antenna. The path loss is set as 128.1 + 37.6 log10 dfor macro cell and 140.7 + 36.7 log10 d for pico cell, respec-tively, where d is the distance in km [19]. A penetration lossof 20 dB is considered for the channels to PUEs. We modelthe interference from the MBSs in adjacent macro cells andthe surrounding PBSs as noises. Define the average receiveSNR of a MUE located at the cell edge as SNRedge, then thenoise variance σ 2

n can be obtained as σ 2n = PM − (128.1 +

37.6 log10 RM ) − SNRedge in dBm. To evaluate the impact ofimperfect self-interference cancellation for FD, we define thesignal to self-interference ratio as SIRself = P0 − σ 2

I in dBin order to reflect the level of self-interference cancellation.The Rayleigh flat small-scale fading channels are considered.The fairness factors are set as αk = 1

K Pin multi-user case.

All the results are averaged over 1000 channel realizations.

B. Narrowband Single-User Case

We first simulate the single-user case, where the MBS servesone MUE, each PBS serves one PUE, and each PBS has Nr = 1FD receive antenna.

Under the assumption of perfect self-interference cancella-tion, the average sum rate achieved by the fICIC is depictedin Fig. 3. For comparison, the performance of the HD scheme,given by E{log(1 + SINR�

k,H D)}, and the performance in ICI


50 100 150 200 250 300 350 400 450 5000

2

4

6

8

10

12

dP 1 (m)

Ave

rage

sum

rat

e (b

ps/H

z)

10

Fig. 4. Average sum rate versus dP1 for different PBS placements, where per-fect self-interference cancellation and perfect channels are considered, KM =1, K P = 1, and Nr = 1.

free case, given by E{log(1 + P0‖hPk‖2

σ 2n

)}, are also presented.

First, we can see that all the schemes perform closely in lowSNR regime, where the system operates in noise-limited sce-nario. With the increase of SNR, the performance floor appearsfor the HD scheme, which is caused by the ICI from the MBS.The performance of Cell 2 is better than Cell 1 due to dP2 >

dP1 that leads to weaker ICI. The proposed fICIC exhibits anoticeable performance gain over the HD scheme. For weakICI case, e.g., Cell 2, as we analyzed, the fICIC can thoroughlyeliminate the ICI in a high probability (considering the random-ness of small-scale channels), and thus the performance gapbetween the fICIC and the ICI free case is very small.

In Fig. 4, we evaluate the impact of the strength of the ICIon the performance of the fICIC by simulating a single PBSwith different positions dP1. The performance achieved by thefICIC and the HD scheme as well as the performance gain ofthe fICIC are depicted. It is shown that the performance of theHD scheme increases slowly with the reduction of ICI, i.e.,the increase of dP1, while the performance of the fICIC firstincreases fast and then keeps nearly constant. It implies thatthe fICIC can make a large area of the macro cell, e.g., from150 m to 500 m, experience very weak ICI. Moreover, we cansee that the fICIC performs close to the HD scheme at largedP1 because the ICI is very weak and has negligible impact onthe performance in this case. Further recalling that the fICICwill degenerate to the HD scheme when the ICI is very strongas analyzed in Section III-B, we can understand that the gain ofthe fICIC over the HD scheme first increases and then decreaseswith dP1. However, we can still observe an evident performanceimprovement of nearly 300% even when dP1 = 10 m, in whichcase the average power of the ICI is 10.4 dB stronger than thatof the desired signals.

Figure 5 shows the performance of the fICIC as a func-tion of SIRself, where imperfect self-interference cancellationis considered. First, it can be seen that the fICIC reduces tothe HD scheme for low SIRself, which agrees with our previousanalysis. Second, the fICIC outperforms the HD scheme whenSIRself ≥ 60 dB for Cell 1 but 75 dB for Cell 2. This is becausePBS1 is closer to the MBS and hence can listen a stronger ICI,which can relax the requirement of self-interference cancella-tion for FD. Moreover, it is shown that the self-interference

Fig. 5. Average sum rate versus SIRself with imperfect self-interference can-cellation, where perfect channels are considered, SNRedge = 20 dB, KM = 1,K P = 1, and Nr = 1.

cancellation of 110 dB is sufficient for the fICIC, which ispractically possible because on one hand existing work hasreported 90 dB self-interference cancellation even for closelyplaced transmit and receive antennas [6], and on the other hand,as we discussed before, in our case the transmit antennas andFD receive antennas can be separated far apart, leading to fur-ther reduction of the self-interference, e.g., with an additional20 dB penetration loss.

C. Narrowband Multi-User Case

In this subsection the multi-user case is simulated, whereeach PBS serves two PUEs. We compare the performance of thefICIC with the HD scheme, a successive interference cancella-tion (SIC) based non-linear HD scheme (denoted by HD-SIC)[20], the ABS-based eICIC and CoMP-CB, and also evaluatethe performance of the combinations of the fICIC with eICICand CoMP-CB. To study the effectiveness of the seven schemesfor cancelling different-strength ICI, we fix the location ofCell 2 and adjust the location of Cell 1 to generate differentinterference to noise ratio (INR) for PUEs in Cell 1, where

INR � E{‖hMk‖2}σ 2

n. To enable the HD-SIC scheme, we consider

that the MBS serves a single MUE because each PUE has onlyone antenna so that only one interference signal can be decodedand cancelled.

In the simulations, the proposed low-complexity algorithmgiven in Table II is employed to obtain the performance of boththe fICIC and the HD scheme, where W f is set as zero for theHD scheme. For the HD-SIC, since the ICI needs to be decodedfirst by regarding the desired signal from the PBS as interfer-ence, the PBS may not transmit with its maximal power. Giventhe data rate of the MUE as 4 bps/Hz, we employ exhaustivesearching to find the maximal feasible transmit power of thePBS under the ICI decoding constraint, where for any giventransmit power the precoder of the PBS is computed the same asthe HD scheme. For the ABS-based eICIC, it is considered thatthe MBS mutes in half time to provide ICI-free environmentto the PUEs. For CoMP-CB, the signal-to-leakage-plus-noiseratio (SLNR) based precoder [21] is employed. The results aredepicted in Fig. 6.

Compared with the fICIC, we can see that the eICIC achievesworse performance when the ICI is not strong, say INR <


Fig. 6. Average sum rate of PUEs in Cell 1 achieved by relevant schemes ver-sus INR, where perfect self-interference cancellation and perfect channels areconsidered, SNRedge = 20 dB, Nr = 2, KM = 1, and K P = 2.

25 dB, because the fICIC can effectively cancel the weak-medium level of ICI and allow the PUEs to use all the time-frequency resources, while with the eICIC the PUEs only useshalf resources. When the ICI is strong, as we analyzed before,the fICIC is not effective, and the eICIC shows large perfor-mance gain. By combining eICIC with fICIC, the two schemessupplement each other and achieve much better performance.

For CoMP-CB, we can see that it is superior to the fICICwhen the ICI is strong, say INR ≥ 26 dB, but inferior for weak-medium ICI. This can be explained as follows. In the simulatedcase, the MBS has only four antennas, which are not adequateto serve one MUE and suppress the ICI for four PUEs simulta-neously. As a result, when the INR of Cell 1 is smaller than Cell2, the SLNR-based CoMP-CB only suppresses the stronger ICIto the PUEs in Cell 2, while the ICI to PUEs in Cell 1 will bemitigated only for large INR (this also explains why the per-formance of Cell 1 with pure CoMP-CB or the combination ofCoMP-CB and fICIC first deceases and then increases with thegrowth of INR.). By contrast, the fICIC is effective to suppressweak-medium ICI but not for strong ICI. Therefore, CoMP-CBand fICIC perform better in different ICI cases. Since CoMP-CB has no enough antenna resources to cancel weak-mediumICI in the simulation, the combination of CoMP-CB and fICIChas negligible gain over the pure fICIC for weak-medium INR.However, evident performance improvement is achieved bytheir combination for strong ICI, because the strong ICI can befirst suppressed into weak-medium ICI by CoMP-CB and thenfurther effectively cancelled by the fICIC.

For the HD-SIC, we can observe that it is not always fea-sible when the ICI is not strong, e.g., INR < 15 dB. With theincrease of ICI, the interference signal becomes decodable andmore power can be transmitted by the PBS, which results in theperformance improvement as expected. It can be seen that theHD-SIC outperforms the fICIC for strong ICI, which inspiresus to investigate the SIC-based fICIC in future work, where theFD PBS forwards the listened ICI to further enhance the ICI atthe PUE so as to extend the feasible region of SIC.

Finally, in Fig. 7 the performance of the fICIC under imper-fect self-interference cancellation, imperfect channel estimationand practical channel feedback is evaluated, where only Cell

Fig. 7. Average sum rate of PUEs in Cell 1 versus SNRedge, where PBS1is uniformly placed within the area suffering from strong ICI with dP1 ∼U ([50, 250])m, SIRself = 110 dB, Nr = 1, 2, KM = 1, 2, and K P = 1, 2. Inthe legends, “p-CE” and “i-CE” denote perfect and imperfect channel esti-mation, respectively, and “p-FB” and “i-FB” denote perfect and imperfectfeedback, respectively.

1 is considered and PBS1 is randomly placed within the areaexperiencing strong ICI, specifically with dP1 following uni-form distribution between [50, 250]m. As discussed before, thechannel HM P can be directly estimated at the PBS, the chan-nel hPk can be obtained at the PBS by estimating the uplinkchannel based on channel reciprocity, and the channel hMk canbe first estimated at the PUEk and then fed back to the PBS.In simulations, we employ linear minimum mean-squared errorestimator to estimate HM P , hPk and hMk , and use analog feed-back [18] to send back the estimate of hMk to the PBS, wherethe transmit power of PUEk is set as 23 dBm. Figs. 7(a) and 7(b)show the results in single-user and multi-user cases, respec-tively. In both cases we can see the performance degradation ofthe fICIC caused by imperfect channel estimation and imperfectfeedback. However, compared with the HD scheme, significantperformance gain achieved by the fICIC can be still observed,even when imperfect self-interference cancellation, imperfectchannel estimation and practical analog feedback are taken intoaccount.

D. Wideband Single-User Case

In this subsection we evaluate the performance of the fICICin wideband systems. We consider a LTE system with 5 MHzbandwidth, where the sampling interval is Ts = 0.13 µs [22].The small-scale channels are generated based on WINNER IIclustered delay line model [23]. Specifically, the channels fromthe MBS to the PBS, HM P , use the typical urban macro-cellline of sight (LoS) model considering that the receive anten-nas of the PBS can be mounted outside a building as discussedbefore, the channels from the MBS to PUEs, hMk , use the typ-ical urban macro-cell non-LoS model, and the channels fromthe PBS to PUEs, hPk , use the typical urban micro-cell NLoSmodel. After sampling the multipath channels with the consid-ered bandwidth, we obtain the maximal delay spread of HM P

and hPk as three and six samples, respectively. We consider theprocessing delay of the FD PBS as 4 samples, i.e., τ = 0.52 µs.


Fig. 8. Average sum rate of the PUE over N = 25 subcarriers in Cell 1 versusSNRedge, where SIRself = 110 dB, KM = 1, and K P = 1.

Since the CP of the LTE system is 4.7 µs [22], i.e., 36 samples,we can obtain the maximal order of the FIR forwarding pre-coder W f (t) as 23, i.e., L ≤ 23. Note that the maximum valueof L can be even larger if the extended CP with 16.7 µs isconsidered.

Considering that the complexity of solving problem (59)increases rapidly with the number of subcarriers and also con-sidering the channel correlation in frequency domain, in thesimulations we treat each resource block (RB) as a subcar-rier and then N = 25 subcarriers are simulated correspondingto the total 25 RBs of the 5 MHz LTE system [22]. In addi-tion, we select the order of W f as L = 1, 2, 4 to speed upthe simulations. In Fig. 8, the average sum rate of Cell 1 withdP1 = 60 m achieved by the fICIC is depicted, where imperfectself-interference cancellation, imperfect channel estimation andpractical channel feedback as considered in Fig. 7 are taken intoaccount. The compared HD scheme is obtained from problem(59) by setting W f as zeros.

We can see from Fig. 8 that the performance of the widebandfICIC improves with the increase of L , and an evident perfor-mance gain over the HD scheme can be observed when L = 4.More significant performance gain can be expected when largerL is used for the fICIC, to achieve which more efficient low-complexity algorithms to problem (59) need to be studied infuture work.

VIII. CONCLUSIONS

In this paper we proposed to eliminate the cross-tier ICIin HetNets by using FD technique at the PBS. We deriveda FD assisted ICI cancellation (fICIC) scheme, with whichthe ICI can be mitigated by merely designing the precodersat PBSs without relying on the participation of the MBS.We first investigated the narrowband single-user case to gainsome insight into the behavior of the fICIC, where we foundclosed-from solution of the optimal fICIC, and analyzed itsasymptotical performance in ICI-dominated scenario. We thenstudied the general narrowband multi-user case, and deviseda low-complexity algorithm to find the optimal fICIC schemethat maximizes the downlink sum rate of PUEs subject to

the fairness constraint. Finally, we generalized the fICIC towideband systems and discussed the practical issues regard-ing the application of the fICIC. Simulations validated theanalytical results. Compared with the traditional HD scheme,the fICIC exhibits significant performance gain even whenimperfect self-interference cancellation and imperfect channelinformation are taken into account. By combining the fICICwith eICIC or CoMP-CB, the ICI with various levels can beeffectively eliminated.

APPENDIX ADERIVATION OF TRANSMIT POWER Pout

By substituting (8) into (9), we can obtain

xp[t] = W f

(HH

M P sM [t − τ ] − EHP P [t − τ ]xp[t − τ ]

+ HHP P zx [t − τ ] + np[t − τ ] + zy[t − τ ]

)+

K P∑k=1

wd,ksp,k[t].

(A.1)

Since the terms at the right-hand side of (A.1) are indepen-dent, we can obtain Pout with (10) by taking the expectationsover each term, yielding

Pout = tr(

W f HHM P HM P WH

f

)+ σ 2

n tr(

W f WHf

)+∑K P

k=1‖wd,k‖2 + tr(

W f �1WHf

)+ tr

(W f �2WH

f

)+ tr

(W f �3WH

f

), (A.2)

where �1 � Exp,EP P ,HP P {EHP P [t − τ ]xp[t − τ ]xH

p [t − τ ]EP P

[t − τ ]}, �2 � EHP P {HHP Pμx diag(�x )HP P }, and �3 �

EHP P {μydiag(�y)}, which can be derived as follows.Denoting xp = [x p1, . . . , x pNt ]

T , we can rewrite �1 as

�1 = Exp,EP P ,HP P

{ Nt∑i=1

|x pi [t − τ ]|2eP P,i eHP P,i

}

(a)= EHP P

{ Nt∑i=1

[�x ]i i �e,i

}

(b)= EHP P

⎧⎨⎩

Nt∑i=1

[�x ]i i

⎛⎝HH

P P

Nt∑j=1

|ci j |2μx diag(c j cHj )HP P

+ 1

Ptr(1 + μy)σ

2n INr +

Nt∑j=1

|ci j |2μy ·

diag(

HHP P

(c j cH

j + μx diag(c j cHj ))

HP P

))⎫⎬⎭

(c)=Nt∑

i=1

[�x ]i i

⎛⎝ Nt∑

j=1

|c2i j |μx αP Ptr

(diag(c j cH

j ))

INr

+ 1

Ptr(1 + μy)σ

2n INr +

Nt∑j=1

|ci j |2μy αP P

(tr(c j cH

j )INr


+μx tr(diag(c j cH

j )INr

))⎞⎠(d)=(

(1 + μy)σ2n

Ptr+ αP P (μx + μy + μxμy)

)tr(�x )INr ,

(A.3)

where the expectations over xp and EP P are taken in step(a), [�x ]i i denotes the i-th diagonal element of �x , step(b) comes from (7), step (c) takes the expectation over HP P

considering vec(HP P ) ∼ CN(0Nr Nt , αP P INr Nt ), and step (d)

comes from the fact that CCH = INt such that tr(ci cHi ) =

tr(diag(ci cHi )) = ∑Nt

j=1 |ci j |2 = 1.The term �2 can be obtained as

�2 = EHP P {HHP Pμx diag(�x )HP P } = αP Pμx tr(�x )INr .

(A.4)

With (3), the term �3 can be obtained as

�3 = EHP P {μydiag(�y)}= EHP P

{μydiag

(HH

M P HM P

+ HHP P (�x + μx diag(�x ))HP P + σ 2

n INr

)}(A.5)

= μy

(diag(HH

M P HM P )+αP P (1+μx )tr(�x )INr +σ 2n INr

).

Substituting (A.3), (A.4) and (A.5) into (A.2) and noting thattr(�x ) = Pout , we can obtain

Pout = tr(

W f

(HH

M P HM P + μydiag(HHM P HM P )

)WH

f

)

+K P∑k=1

‖wd,k‖2 +(

(1 + μy)σ2n

+((1+μy)σ

2n

Ptr+2αP P (μx+μy+μxμy)

)Pout

)tr(W f WH

f )

≈ tr(

W f HHM P HM P WH

f

)+

K P∑k=1

‖wd,k‖2+(

σ 2n +

(σ 2

n

Ptr+ 2αP P (μx + μy)

)Pout

)tr(W f WH

f ), (A.6)

where the approximation follows from μx 1 and μy 1 asin [8].

APPENDIX BPROOF OF LEMMA 1

To prove this lemma, we rewrite problem (21) by express-ing W f with its vectorization, denoted by w f = vec(W f ), asfollows.

maxw f ,wd,k

|hHPkwd,k |2

�′k

(B.1a)

s.t. ‖(

hTM P ⊗ INt

)w f ‖2+(P0σ

2e +σ 2

n )‖w f ‖2+‖wd,k‖2≤P0,

(B.1b)

where �′k � |h�

Mk + e− jφ(hT

M P ⊗ hHPk

)w f |2+‖ (INr ⊗hH

Pk

)w f ‖2(P0σ

2e + σ 2

n ) + σ 2n , and the property vec(AXB) =(

BT ⊗ A)

vec(X) is used.

Based on the KKT condition, we can obtain the optimalsolution of w f as

w�f = −h�

Mke jφ(

SINRk,F D

�1

(hT

M P ⊗ hHPk

)H (hT

M P ⊗ hHPk

)+SINRk,F D�2

�1

(INr ⊗hH

Pk

)H(INr ⊗hH

Pk

)+λ(hT

M P ⊗ INt

)H

·(

hTM P ⊗ INt

)+ λ�2INt Nr

)−1 (hT

M P ⊗ hHPk

)H, (B.2)

where �1 = |h�Mk + e− jφ

(hT

M P⊗hHPk

)w�

f |2+‖(INr ⊗hHPk

)w�

f ‖2

(P0σ2e + σ 2

n ) + σ 2n , �2 = P0σ

2e + σ 2

n , and λ is the lagrangianmultiplier.

By applying the properties of Kronecker product and aftersome regular manipulations, we can simplify (B.2) as

w�f = − h�

Mke jφ((

(hM P hHM P )T + �2INr

)−1(hH

M P )T)

⊗((

SINRk,F D

�1hPkhH

Pk + λINt

)−1

hPk

). (B.3)

Then based on the matrix inversion lemma, we can furtherrewrite (B.3) as

w�f = −h�

Mke jφ�1(hHM P )T ⊗ hPk(

SINRk,F D‖hPk‖2 + λ�1) (‖hM P‖2 + �2

)� −h�

Mke jφ · β · (hHM P )T ⊗ hPk, (B.4)

where β = �1(SINRk,F D‖hPk‖2+λ�1)(‖hM P‖2+�2)

is a positive

scalar.According to (B.4), we can recover the optimal W�

f from w�f

as (24), which is of rank 1.

APPENDIX CPROOF OF PROPOSITION 1

By defining ν � |hMk |‖hPk‖‖hM P‖β = C2 β, problem (25)

can be rewritten as

maxν

f (ν) � A − Bν2

Bν2 − Cν + D(C.1a)

s. t. ν2 ≤ A

B(C.1b)

ν ≥ 0, (C.1c)

where the parameters A, B, C and D are defined inProposition 1.

Since the numerator of the objective function (C.1a) is con-cave, the denominator is convex, and both are differential, weknow that the objective function of problem (C.1) is quasi-concave [14]. This suggests that the global optimal solutionto problem (C.1), denoted by ν, needs to satisfy the followingKKT conditions,

−∇ f (ν) + 2uν − v = 0 (C.2a)

u(ν2 − A

B) = 0, u ≥ 0 (C.2b)

vν = 0, v ≥ 0, (C.2c)


where u and v are the lagrangian multipliers.We next show that u = 0 and v = 0 for problem (C.1). First,

we find that ν2 < AB must hold for the constraint (C.1b), oth-

erwise, when ν2 = AB , the objective function will be zero that

is obviously non-optimal. Therefore, from (C.2b) we have u =0. Second, the objective function satisfies ∇ f (0) > 0, whichmeans that we can always find a small ε > 0 making f (ε) >

f (0), i.e., ν > 0 must hold. Therefore, based on (C.2c) we havev = 0.

Then the condition (C.2a) for ν can be simplified as

2Bν − (Bν2 − A)(2Bν − C)

Bν2 − C ν + D= 0. (C.3)

From (C.3), we can derive the optimal value of the objectivefunction as

f (ν) = 2Bν

C − 2Bν= 1

C2Bν

− 1. (C.4)

The condition (C.3) can be further expressed as a quadraticequation with respect to ν, which is

g(ν) � BC ν2 − 2B(A + D)ν + AC = 0, (C.5)

whose two solutions can be obtained as

ν = A + D ±√

(A + D)2 − AC2

B

C. (C.6)

The feasibility of the two solutions is examined as follows.

Recalling that 0 ≤ ν ≤√

AB from (C.1b) and (C.1c), we can see

that g(0) = AC > 0 and g(

√AB ) < 0 because

g

(√A

B

)= 2AC − 2 (A + D)

√AB

< 4P0|hMk |‖hM P‖‖hPk‖3 − 2(P0‖hPk‖2 + |hMk |2)·√P0‖hM P‖2‖hPk‖4

= −2√

P0‖hM P‖‖hPk‖2(|hMk | −√P0‖hPk‖)2, (C.7)

where the inequality is obtained by setting σ 2e = 0 and σ 2

n = 0in the definitions of B and D.

The results indicate that equation (C.5) has one and only

one solution within [0,

√AB ], which is the smaller one in (C.6).

Finally, recalling that ν = C2 β, we can obtain the optimal β as

shown after (26).

APPENDIX DPROOF OF PROPOSITION 2

By defining A = INt +∑j �=k λ j hP j hH

P j and B = INt −λkγk

A− 12 hPkhH

PkA− H2 , we can express the left-hand side of (44b)

as A12 BA

H2 . Since A is positive definite and B is Hermitian,

it is not difficult to show that A12 BA

H2 � 0Nt and B � 0Nt

are equivalent to each other. Then, the semi-definite positive

constraints (44b) can be equivalently expressed as

INt − λk

γk

⎛⎝INt +

∑j �=k

λ j hP j hHP j

⎞⎠− 1

2

hPkhHPk

×⎛⎝INt +

∑j �=k

λ j hP j hHP j

⎞⎠− H

2

� 0Nt ,∀k. (D.1)

The positive semi-definite constraint in (D.1) means thatthe minimal eigenvalue of the matrix in the left-hand

side should be non-negative. Note that the term(

INt +∑j �=k λ j hP j hH

P j

)− 12

hPkhHPk

(INt +∑

j �=k λ j hP j hHP j

)− H2

is

of rank one. It has only one positive eigenvalue, which is

hHPk

(INt +∑

j �=k λ j hP j hHP j

)−1hPk , and all other eigenvalues

are zeros. Therefore, the constraint in (D.1) can be further con-verted into the constraint on the minimal eigenvalue of thematrix in the left-hand side of (D.1) as

1 − λk

γkhH

Pk

⎛⎝INt +

∑j �=k

λ j hP j hHP j

⎞⎠−1

hPk ≥ 0,∀k. (D.2)

which can be further rewritten as (45).

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Shengqian Han (S’05–M’12) received the B.S.degree in communication engineering and the Ph.D.degree in signal and information processing fromBeihang University (formerly Beijing University ofAeronautics and Astronautics), Beijing, China, in2004 and 2010, respectively. From 2010 to 2012,he held a postdoctoral Research position with theSchool of Electronics and Information Engineering,Beihang University, where he is currently a Lecturer.His research interests include cooperative commu-nication, full-duplex communications, and energy

efficient transmission in the areas of wireless communications and signalprocessing.

Chenyang Yang (SM’08) received the M.S.E.and Ph.D. degrees in electrical engineering fromBeihang University (formerly Beijing University ofAeronautics and Astronautics), Beijing, China, in1989 and 1997, respectively. She is currently aFull Professor with the School of Electronics andInformation Engineering, Beihang University. Shehas authored various papers and filed many patentsin the fields of signal processing and wireless com-munications. Her research interests include signalprocessing in network MIMO, cooperative commu-

nication, energy efficient transmission, and interference management. She wasthe Chair of the IEEE Communications Society Beijing Chapter from 2008to 2012. She has served as TECHNICAL PROGRAM COMMITTEE MEMBER

for many IEEE conferences, such as the IEEE International Conference onCommunications and the IEEE Global Telecommunications Conference. Shecurrently serves as an Associate Editor for the IEEE TRANSACTIONS ON

WIRELESS COMMUNICATIONS, an Associate Editor-in-Chief of the ChineseJournal of Communications, and an Associate Editor-in-Chief of the ChineseJournal of Signal Processing. She was nominated as an Outstanding YoungProfessor of Beijing in 1995 and was supported by the First Teaching andResearch Award Program for Outstanding Young Teachers of Higher EducationInstitutions by Ministry of Education (P.R.C.TRAPOYT) from 1999 to 2004.

Pan Chen received the B.S. degree in electron-ics engineering from Beihang University (formerlyBeijing University of Aeronautics and Astronautics),Beijing, China, in 2013. She is currently pursuingthe M.S. degree in signal and information process-ing at the School of Electronics and InformationEngineering, Beihang University. Her research inter-ests include cellular interference management andfull duplex communications.

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